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Mirrors > Home > ILE Home > Th. List > opid | GIF version |
Description: The ordered pair 〈𝐴, 𝐴〉 in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.) |
Ref | Expression |
---|---|
opid.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
opid | ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsn2 3541 | . . . 4 ⊢ {𝐴} = {𝐴, 𝐴} | |
2 | 1 | eqcomi 2143 | . . 3 ⊢ {𝐴, 𝐴} = {𝐴} |
3 | 2 | preq2i 3604 | . 2 ⊢ {{𝐴}, {𝐴, 𝐴}} = {{𝐴}, {𝐴}} |
4 | opid.1 | . . 3 ⊢ 𝐴 ∈ V | |
5 | 4, 4 | dfop 3704 | . 2 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}, {𝐴, 𝐴}} |
6 | dfsn2 3541 | . 2 ⊢ {{𝐴}} = {{𝐴}, {𝐴}} | |
7 | 3, 5, 6 | 3eqtr4i 2170 | 1 ⊢ 〈𝐴, 𝐴〉 = {{𝐴}} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 {csn 3527 {cpr 3528 〈cop 3530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 |
This theorem is referenced by: dmsnsnsng 5016 funopg 5157 |
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